The Parity Problem for Reducible Cubic Forms

Helfgott, Harald Andrés

doi:10.1112/s0024610706022629

Cited by 14 publications

(13 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We remark that this conjecture was also recently verified for all homogeneous polynomials of degree at most three in [27], [28]. Removing the homogeneity assumption looks hopeless with current technology; the case P .y 1 ; y 2 / D y 1 .y 1 C 2/ is already roughly of the same order of difficulty as the twin prime conjecture.…”

Section: Correlation Estimates For Möbius and Liouvillesupporting

confidence: 56%

Linear equations in primes

2010

View full text Add to dashboard Cite

Consider a system ‰ of nonconstant affine-linear forms 1 ; : : : ; t W ‫ޚ‬ d ! ‫,ޚ‬ no two of which are linearly dependent. Let N be a large integer, and let K Â OE N; N d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N ! 1, for the number of integer points n 2 ‫ޚ‬ d \ K for which the integers 1 .n/; : : : ; t .n/ are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime.In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms 1 ; : : : ; t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI.s/) and the Möbius and nilsequences conjecture (MN.s/), where s 2 f1; 2; : : : g is the complexity of the system and measures the extent to which the forms i depend on each other. The case s D 0 is somewhat degenerate, and follows from the prime number theorem in APs.Roughly speaking, the inverse Gowers-norm conjecture GI.s/ asserts the Gowers U sC1 -norm of a function f W OEN ! OE 1; 1 is large if and only if f correlates with an s-step nilsequence, while the Möbius and nilsequences conjecture MN.s/ asserts that the Möbius function is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s D 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s D 2 in two papers of the authors. Thus our results in the case of complexity s 6 2 are unconditional.In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 6 N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns. 1753 1754 BENJAMIN GREEN and TERENCE TAO

show abstract

Section: Correlation Estimates For Möbius and Liouvillesupporting

confidence: 56%

Linear equations in primes

2010

View full text Add to dashboard Cite

show abstract

“…Helfgott has shown that the root number in this family is equidistributed [He2]. We have the following.…”

Section: Curves With Prescribed Torsionmentioning

confidence: 87%

Low-lying zeros of families of elliptic curves

Young

2005

J. Amer. Math. Soc.

108

View full text Add to dashboard Cite

There is a growing body of work giving strong evidence that zeros of families of L L -functions follow distribution laws of eigenvalues of random matrices. This philosophy is known as the random matrix model or the Katz-Sarnak philosophy. The random matrix model makes predictions for the average distribution of zeros near the central point for families of L L -functions. We study these low-lying zeros for families of elliptic curve L L -functions. For these L L -functions there is special arithmetic interest in any zeros at the central point (by the conjecture of Birch and Swinnerton-Dyer and the impressive partial results towards resolving the conjecture). We calculate the density of the low-lying zeros for various families of elliptic curves. Our main foci are the family of all elliptic curves and a large family with positive rank. An important challenge has been to obtain results with test functions that are concentrated close to the origin since the central point is a location of great arithmetical interest. An application of our results is an improvement on the upper bound of the average rank of the family of all elliptic curves (conditional on the Generalized Riemann Hypothesis (GRH)). The upper bound obtained is less than 2 2 , which shows that a positive proportion of curves in the family have algebraic rank equal to analytic rank and finite Tate-Shafarevich group. We show that there is an extra contribution to the density of the low-lying zeros from the family with positive rank (presumably from the “extra" zero at the central point).

show abstract

“…A variant of Chowla's conjecture. A classical conjecture of Chowla [9] states that if λ is the Liouville function and P ∈ Z[x, y] is a homogeneous polynomial such that P = cQ 2 for every c ∈ Z, Q ∈ Z[x, y], then This was established by Landau when deg(P ) = 2 [52] (see also [39]), by Helfgott when deg(P ) = 3 [40,41], and when P is a product of pairwise independent linear forms by Green, Tao, and Ziegler [29,30,31,32]. The conjecture is also closely related to the problem of representing primes by irreducible polynomials, for relevant work see [16,36,37,38].…”

mentioning

confidence: 99%

Higher order Fourier analysis of multiplicative functions and applications

Frantzikinakis

Host

2016

J. Amer. Math. Soc.

161

View full text Add to dashboard Cite

We prove a structure theorem for multiplicative functions which states that an arbitrary multiplicative function of modulus at most 1 can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and finitary ergodic theory, and some soft number theoretic input that comes in the form of an orthogonality criterion of Kátai. We use variants of this structure theorem to derive applications of number theoretic and combinatorial flavor: (i) we give simple necessary and sufficient conditions for the Gowers norms (over N) of a bounded multiplicative function to be zero, (ii) generalizing a classical result of Daboussi we prove asymptotic orthogonality of multiplicative functions to "irrational" nilsequences, (iii) we prove that for certain polynomials in two variables all "aperiodic" multiplicative functions satisfy Chowla's zero mean conjecture, (iv) we give the first partition regularity results for homogeneous quadratic equations in three variables, showing for example that on every partition of the integers into finitely many cells there exist distinct x, y belonging to the same cell and λ ∈ N such that 16x 2 + 9y 2 = λ 2 and the same holds for the equation x 2 − xy + y 2 = λ 2 .

show abstract

The Parity Problem for Reducible Cubic Forms

Abstract: Let f ∈ Z[x, y] be a reducible homogeneous polynomial of degree 3. We show that f (x, y) has an even number of prime factors as often as an odd number of prime factors.

Cited by 14 publications

References 21 publications

Linear equations in primes

Linear equations in primes

Low-lying zeros of families of elliptic curves

Higher order Fourier analysis of multiplicative functions and applications

Contact Info

Product

Resources

About